
also satisfies the boundary conditions in (REF ). Then, according to ([1]}, Cor. 3.24), the function \(\bar{u}(t, \chi _{1})\)  is also a continuous solution to the integral equation
\(\bar{u}(t, \chi _{1}) =& \alpha _{1} + \chi _{1}t + \frac{1}{\Gamma (p)}\int _{0}^{t} (t-s)^{p-1}f(s, \bar{u}(s, \chi _{1}) ds + \frac{t^{p}\bar{\mu }}{\Gamma (p+1)}.\) 
